esercizi di analisi matematica 2

8/11/2019 esercizi di analisi matematica 2

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A Giuliacon la speranza che almeno nella matematica

non assomigli al pap


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y(t)

y =

y2

y2 + 4t

y(0) = 2

> 0

y(t)

t

t+

y2 + 4

y2

dy =

t dt+C

y 4y

=

1 +

4

y2

dy=

y2 + 4

y2

dy=

t2

2 +C


8/236

C= 0

y2 4 t2y2

y = 0.

y

y(t) =t2/2

t4

4 + 16

2 =

t2 t4 + 644

.

y(0) =2

y(t) = t2 +

t4 + 64

4 .

t4 + 64

t2

t

+

y(t)

x2

2

= 2

y(t)

y =

t2 +t

2e2y + 6ey

y(0) = 0

e2y + 6ey =

(2e2y + 6ey) dy =

(t2 +t) dt=

t3

3 +

t2

2 +C.

C= 7

e2y + 6ey t3

3t

2

2 7 = 0.

y

z = ey

(z+ 3)2 =z2 + 6z+ 9 = 9 + t3

3 +t

2

2 + 7.

ey =z=3

16 +t3

3 +

t2

2.

y(t) = log

3 +

16 +

t3

3 +

t2

2

.


9/236

y(t)

y = 3ex y2y(0) = 1

y(t)

x0 = 1

y(0) = 3 0 = 3> 0

y(0)> 0

x0 = 1

y = 3ex 2y y

y(0) = 3 2y(0) y(0) = 3 2 1 3 =3< 0.

x0 = 1

y(t)

y =ex

y+ 1

ex + 1y(0) = 1


10/236

y= 1

2log(ex + 1) +

2 +

1

2log2

2

1.

y(t)

y = (e3y + 1)(2x 1)y(0) =1

y(t) =1

3log

(1 +e3)e3x

23x 1

.

y(t)

y = (3 + 27y2) (xe3x 2x2)y(0) = 0

y(t) =13tan

3t e3t e3t 6t3 + 1 .

y =

2xe3y.


11/236

y(t)

y=

y

t+ 1y(0) = 1

y(1) =

e

2

e

y(t) y

(t+ 2)

1 +1

y

= 0

y(0) = 1

y(t)

y(t)

y = 3 sin t+y2y(0) =

y(t)

x0= 0


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13/236

2 =y (0) = 3c1+c2+1

3.

c1

c1=139

, c2= 2 13 3c1 = 5

3+

13

3 = 6.

y(t) =139

e3t + 6 t e3t +1

3t+

4

9.

y(t)

y+ 2y 3y= 0y(0) = 0

y(0) = 1.

limt+ y(t) =

+

y + 2y3

r2 + 2r 3 = 0

r= 1

r=3

y(t) =c1 et +c2 e

3t,

c1, c2R

0 =y(0) =c1+c2

y(t) =c1 et 3c2 e3t

1 =y (0) =c1 3c2

c1 =1

4; c2=1

4.


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15/236

y(t) = 120

sin(2t) 320

cos(2t).

y(t) =c1 et +c2 e

2t 120

sin(2t) 320

cos(2t).

y(t) =11

15et +

5

12e2t

1

20sin(2t)

3

20cos(2t).

y(t)

y 4y+ 8y=e2ty(0) =1y(0) = 0.

r2 4r+ 8 = 0

r= 2 2i

y(t) =c1 e2t sin(2t) +c2 e

2t cos(2t).

y(t) = A e2t

A= 1/20

y(t) =c1 e2t sin(2t) +c2 e

2t cos(2t) + 1

20e2t.

y(t) =11

10e2t sin(2t) 21

20e2t cos(2t) +

1

20e2t.


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y(t)

y y 2y = sin(2t)y(0) = 0

y(0) = 1.

y(t) = 715

et + 5

12e2t 3

20sin(2t) +

1

20cos(2t).

y4y+13y= 4x.

2y+ 3y+ 4y= 0.

y+ 6y+ 8y = e4t +t2, y(1) = 2, y(1) = 3.

y+ 2(tan t)y=t, y(1) = 4.

y(t)

y+y 2y=exy(0) = 0

y(0) = 0.


17/236

y(t) =1

9et 1

9e2t 1

3t et

y+ 3(cot t)y = t, y(1) =.

y+ 6y+ 8y = e2t

+ t2

, y(1) = 0, y(1) = 2.

y 6(cot t)y= t, y(2) =.

y+ (tan t)y= 2t2, y(1) = 3.

y(3) 2y+ 5y = 0.

y(t)

limt

y(t) =.

y(3) 2y+ 5y = 3tet.


18/236

y

6y+ 9y = 0.

y 6y+ 9y= cos(

2t).

y 4y+ 13y = 0.

y 4y+ 13y= 1 +e2t.

y+y = 0.

y+y= 1 +et.

y 2y+ 17y = 0.

y 2y+ 17y = sin(2t).


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20/236

N() =

2 + 14 + 62 + 1

, 2

4 + 62 + 1

.

x=(2 + 1) 2ty=t(2 + 1) tR

y=2 + 1

2 x+

(2 + 1)2

2 .

f(x) = (1/3)(2x 1)3/2 1/2

x1

f

1=

x= t

y= 13

(2t 1)3/2 12t1;

2= x= 32 12 ty= 1

3(2 t) 1t2.

1(t) = (1,

2t 1).

L() =L(1) + L(2) =

11/2

1 + (2t 1) dt +

1

2

2+

1

3

2

=

1

1/2

2t dt+

136

=

2

2

3t3/2

1

1/2

+13

6 =

223

13

+13

6

= 22

,

;

=

0+


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(t) = ((t)cos t, (t)sin t) = (2t2 cos t, 2t2 sin t)

t,

(t) = (4t cos t 2t2 sin t, 4t sin t+ 2t2 cos t)

|(t)|2 = 16t2 + 4t4 |(t)|=

4t2(4 +t2) = 2|t|

4 +t2,

L=

2|t|4 +t2 dt= 2 0

t2 + 4 2t dt= 43 (t2 + 4)3/2

0=43(

2 + 4)3/2 323 .

()|()| =

(4 cos 22 sin , 4 sin + 22 cos )2||4 +2

>0,

() = 2cos

4 +2

sin 4 +

2,

2sin 4 +

2+

cos 4 +

2(1, 0)

(et cos t, et sin t) 2 t

2,

;

t= 0.

(t) = (et cos t, et sin t)

(t) = (et(cos t sin t), et(sin t+ cos t),

|(t)|= et

cos2 t+ sin2

t 2sin t cos t+ sin2

t+ cos2 t+ 2 sin t cos t= et

2.

L() =

22

et

2 dt=

2(e2 e2).

t= 0

(0) = (1, 0)

(0) = (1, 1),

x= 1 +t

y=t


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23/236

R3

x= t e2t, y= 3 t e2t, z= 2 t e2t t

[0, 1].

t= 1/2.

(t) := (x(t), y(t)) =

5t33

,5t22 1 , t0.

A = (0),

B = (s),

AB

5(

8 1)3

.

B.

(t) := (x(t), y(t), z(t)) = (et cos t, et sin t, et

3), t(, +).

t0 (t0)

t1 (t0) (t1) 2

5.

(t)

(t1).

(t) =

cos2t+ 2t sin2t, sin2t 2t cos2t,2

3t3

t[0, +).

t >0

(0)

(t)

38

3 .

2

.


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ex

x= 0

x = 1;

f(x, y) = yex;

.

=

x(t) =t t[0, 1]y(t) =et;

(t) = (1, et).

f(x, y)ds= 1

0et et1 +e2t =

1

2(1 +e2t)3/2

3/21

0=

1

3[(1 +e2)3/2 1].

L() =

10

1 +e2t dt.

1 +e2t dt

1 +e2t =:z

2t= log(z2 1).

1 +e2 t dt= z zz2 1dz= 1 + 1z2 1 dz= z+12log z 1z+ 1 +C.

L() =

10

1 +e2t dt=

1 +e2t +

1

2log

1 +e2t 11 +e2t + 1

10

=

1 +e2 +1

2log

1 +e2 11 +e2 + 1

2 1

2log

2 12 + 1

.


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(t) = (t cos2t, t sin2t),

t = 0

f(x, y) =

x2

+y2

1t1.

(t) = (cos 2t2t sin2t, sin2t2t cos2t),

(0) = (1, 0);

(0) = (0, 0)

(0, 0)

(1, 0),

x.

|(t)|2 = cos2 2t+ 4t2 sin2 2t 4t sin2t cos2t+ sin2 2t+ 4t2 cos2 2t+ 4t= 1 + 4t2

|(t)|= 1 + 4t2.

11

t2 cos2 2t+t2 sin2 2t |(t)| dt=

11

t2

1 + 4t2 dt=

11

|t|

1 + 4t2 dt

=2

1

0

1 + 4t2 t dt=

1

4 5

1

s ds=

1

4

2

3[s

s]51=

1

6(5

5 1).

(sin t,t, 1)

0 t 2;

t= 0, t= 2

, t=

,

xyz

1 + cos2 y.

sin x z = 1. (t) = (cos t, 1, 0) (0) = (1, 1, 0)(/2) = (0, 1, 0) () = (1, 1, 0).

xyz

1 + cos2 y ds=

20

sin t t (1 + cos2 t) dt=

20

t sin t dt+

20

t sin t cos2 t dt

=[t cos t+ sin t]20 +tcos

3 t

3 +

1

3sin t sin

3 t

9

20

=83

.


26/236

f(x, y) = xy

4 +x2

E

E=

(x, y) :x0, x2 +y2 1, 0y1 x

2

4

1,

3

4

.

f(x, y) = 1

f(x, y) + 2

f(x, y) + 3

f(x, y)

1=

x= t

y= 0 1t2;1

f(x, y) = 0

2=

x= cos t

y= sin t 0t 2

;

2

f(x, y) =

/20

sin t cos t4 + cos2 t

dt= [

4 + cos2 t]/20 = [2 +

5] =

5 2

3 =

x= t

y= 1 t2

4 0t2;

(x)2 + (y)2 =

1 +

t2

4 =

4 +t2

2 .

3

f(x, y) =

20

t(1 t24)

4 +t2

4 +t2

2 dt=

1

2

20

t t

3

4

dt=

1

2.

f(x, y) =

5 32

.


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1,

3

4

x= t

y= 1 t2

4 0t2;

x= 1, y = t

2

1, 3

4

t= 1

x= 1 +t

y=3

41

2t t R;

y=12x+54 .

f(x, y) =

x y

(x, y) R2 : x2

9 +

y2

4 = 1

s= sin t . . .

() =

x= 3 cos

y= 2 sin

0,

2

() =

x=3 sin

y = 2 cos

0,

2

|()|=

9sin2 + 4 cos2 =

5sin2 + 4.

f ds=

/20

f(()) |()| ds= /2

0

6 sin cos

5sin2 + 4 d

=3

5

/20

10 sin cos

5sin2 + 4 d=

3

5

(5 sin2 + 4)3/2

3/2

/20

=38

5 .


28/236

z ds,

(t) = (t cos t, t sin t, t), t [0, 2].

(, 0, )

(t) := (1(t), 2(t), 3(t)) = (t cos t, t sin t, t) t[0, 2]

(t) = (cos t t sin t, sin t+t cos t, 1)

|(t)|=

cos2 t+t2 sin2 t 2 t sin t cos t+ sin2 t+t2 cos2 t+ 2 t sin t cos t+ 1 = t2 + 2.

f(x,y,z) =z

z ds=

2 0

f(1(t), 2(t), 3(t)) |(t)| dt= 2

0

t

t2 + 2 dt=1

2

(t2 + 2)3/2

3/2

2

0

=1

3[(42 + 2)3/2 23/2].

: R R3

(x0, y0, z0)

(x x0) 1(t0) + (y y0) 2(t0) + (z z0) 3(t0) = 0

(x+) 1() + (y 0) 2() + (z ) 3() = 0

z x y = 2 .

x2 +y2 ds,

(t) = (t sin t, t cos t, 2 t), t[0, 2].

(/2, 0, )

(t) := (1(t), 2(t), 3(t)) = (t sin t, t cos t, 2 t) t[0, 2]


29/236

(t) = (sin t+t cos t, cos t t sin t, 2)

|(t)|=

sin2 t+t2 cos2 t+ 2 t sin t cos t+ cos2 t+t2 sin2 t 2 t sin t cos t+ 4 = t2 + 5.

f(x,y,z) =

x2 +y2

x2 +y2 ds=

2 0

f(1(t), 2(t), 3(t)) |(t)| dt= 2

0

t

t2 + 5 dt=1

2

(t2 + 5)3/2

3/2

2

0

=1

3[(42 + 5)3/2 53/2].

: R R3

(x0, y0, z0)

(x x0) 1(t0) + (y y0) 2(t0) + (z z0) 3(t0) = 0

x 2

1

2

+ (y 0) 2

2

+ (z ) 3

2

= 0

x

2

1 + (y 0) 2 + 2 (z ) = 0

2x y+ 4 z= 5 .

xds,

(t) = (t2, cos t, sin t), t[0, 2].

(2, 1, 0)

(t) := (1(t), 2(t), 3(t)) = (t2, cos t, sin t) t[0, 2]

(t) = (2t, sin t, cos t)

|(t)|= 4t2 + sin2 t+ cos2 t=

4t2 + 1.


30/236

f(x,y,z) =

x

x ds=

2

0

f(1(t), 2(t), 3(t)) |(t)| dt= 2

0

t

4 t2 + 1 dt=1

8

(4 t2 + 1)3/2

3/2

2

0

=1

12[(162 + 1)3/2 1].

: R R3

(x0, y0, z0)

(x x0) 1(t0) + (y y0) 2(t0) + (z z0) 3(t0) = 0

(x

2) 1() + (y+ 1) 2() + (z

0) 3() = 0

(x 2) 2 z= 0

z= 2 x 2 3.


31/236

1)f(x, y) = 3

1 x

2y

4

2)f(x, y) = x2 y2

3)f(x, y) = x y

x+y

f

3

1 x2y

4

= C

x

2+

y

4= 1 C

3 0C3

x2 y2 =C C= 0

x= y

x=y

C


32/236


33/236

f(x, y) = (x2 2x

y)(x2

2x+y)

x 32

2

+

y 12

2

+ lnx+ 1

2 x

(x2 2x y)(x2 2x+y)0x 3

2

2+

y 1

2

2= 0

x+ 12 x>0

x2 + 2xyx2 2x

x=32

, y=12

1< x


34/236

f(x, y) =1 +xy

x2

c

c R

z = f(x, y)

f

c,

Ec={(x, y) R2 : f(x, y) =c}.

c

f

Ec

Ec

1 +xy

x2 =c, c R

y= cx 1x

, cR.

f


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36/236


37/236


38/236

sin z

ez

z

tan z

z=

2+ k

k Z z z0.

f=

(x, y) R2 : x y0 exy =

2+k

k Z .

k Z

e

xy =

2+k

e

xy = 2

+k k Z+ {0}.

xy

= log

2

+k 2 k Z+

{0

}.

f=

(x, y)R2 : xy0

xy=

log

2

+k 2

k Z+ {0}

.

xy 0

k Z+ {0}

xy= log 2

+k 2

f

g,

ez

sin z

z

tan z

z= 2

+k

k Z.

z

z0.

g= (x, y) R

2 : sin(xy)

0

e

x

=

2

+k

k

Z .

k Z e

x =

2+k ,

g=

(x, y) R2 : 2 h x y + 2 h x=

log

2

+ k 2

h, k Z+ {0}

.

y.


39/236

g.

f(x, y) = log(2e2+sin[(xy)]2), g(x, y) =

tan x1 ey .

ez

sin z

z R,

log z

z > 0

z

z0.

log(a b) = log a + log b

log(2 e

2+sin[(xy)]2) = log 2 + loge

2+sin[(xy)]2 = log 2 +

2 + sin[(xy)]2

2 + sin[(xy)]2

0. 1 sin z 1 z R, (x, y) R2. f= R2.

g

ez

z

tan z

z=

2+ k

z

z0.

g=

(x, y) R2 : tan x

1 ey 0 x=

2+k

k Z

.


40/236

k Z

tan x

1 ey 0[tan x0 1 ey >0] [tan x0 1 ey


41/236

f=

(x, y) R2 : 1 |y x| 0

1 |yx| =

2+k

k Z

.

f.

1 |y x| 0 |x y| 1 1x y1

k Z

1 |x y| =

2+k

1 |x y| =

2+ k

k Z+ {0}

|yx| = 1

2+k

2;

k Z+{0}, 1

2

+k 2

|y x| = 1

2+k

2.

f={(x, y) R2 :1x y1}.


42/236

g

z

z0,

g={(x, y) R2 :

1 |y x| 1 0 1 |y x| 0}.

1 |y x| 0 |x y| 1 1x y1.

1 |yx| 1 0

1 |yx| 11 |y x| 1 |y x| 0x = 0 y= 0.

g={(x, y) R2 : x= 0 y = 0}.

f(x, y) = log

1 |yx|, g(x, y) =

|

1 |yx| 1|.

z

z 0,

log z

z > 0.

f

f={(x, y)R2

: 1 |x y|> 0}={(x, y) R2

:1< xy


43/236

1)f(x, y) = xy y22x2 +y2

2)f(x, y) = (x2 +y2) log(1 +x2 +y2)

3)f(x,y,z) =ex2+2y2+3z2

4)f(x,y,z) = log z

x2 +y2

5)f(x,y,z) = arctan x2 +y2

(z 1)2

1)f(x, y) =

(x2 + 4y2 3)2)f(x, y) =

|x|((x+ 1)2 (y 1)2)

3)f(x, y) = x sin(x2 +y2)

4)f(x, y) =

2x y(y |y|)log(1 (x2 +y2))

A B

1)f(x, y) = log(1 x2) log(y2 4)2)g(x, y) = log

1 x2y2 4

A

B

A= B, AB, BA.

f(x, y) = xy2

x ln y


44/236


45/236


46/236

y =x

1

n,1

n+

1

n

>0;

y =x

= 5

1

n,1

n+

1

n5

lim(x,y)(0,0)

x3y3

x3 +y3 = lim

n+

1n3

1

n+

1

n5

33

n7+

3

n11+

1

n15

= limn+

1 3n4

3n8

1n12

3

n+

3

n5+

1

n9

=.

f

(0, 0).

f(x, y) =

x3y

x4 +y2 (x, y)= (0, 0)

0 (x, y) = (0, 0)

(x, y)

= (0, 0).

f

(x, y) (0, 0)

(0, 0).

0 lim(x,y)(0,0)

| x3| |y|x4 +y2

lim(x,y)(0,0)

|x| x4 +y2

2(x4 +y2)= lim

(x,y)(0,0)|x|2

= 0

x2|y| 1

2(x4 +y2)

(x2 + |y|)2 0

|f| 0f0


47/236

f(x, y) =

xy3

x4 +y2 (x, y)= (0, 0)

0 (x, y) = (0, 0)

(x, y)= (0, 0).

f

(x, y) (0, 0)

(0, 0).

0 lim(x,y)(0,0)

| x| |y3|x4 +y2

lim(x,y)(0,0)

| xy| y2

x4 +y2 lim

(x,y)(0,0)| xy|= 0

y2

x4 +y2 1.

|f| 0f0

f(x, y) =

x2y

x4 +y2 (x, y)= (0, 0)

0 (x, y) = (0, 0)

(x, y)= (0, 0).

f

(x, y) (0, 0)

(0, 0).

lim(x,y)(0,0)

x2y

x4 +y2

y=x

lim(x,y)(0,0)

x2y

x4 +y2= lim

x0x3

x4 +x2= lim

x0x

x2 + 1= 0


48/236

y= x2,

lim(x,y)(0,0)

x2y

x4 +y2= lim

x0x4

x4 +x4= lim

x01

2=

1

2.

y= x2

lim(x,y)(0,0)

xy3 2 sin(x2y) cos(x+ 2y)x2 +y2

lim(x,y)(0,0)

xy3

x2 +y2 lim

(x,y)(0,0)2sin(x2y)cos(x+ 2y)

x2 +y2

[+, ].

lim(x,y)(0,0)

xy3

x2 +y2

|x||y| 1

2(x2 +y2),

0 lim(x,y)(0,0)

|x||y3|x2 +y2

12

lim(x,y)(0,0)

y2 = 0.

f0 |f| 0

lim(x,y)(0,0)

xy3

x2 +y2

lim(x,y)(0,0)

2 sin(x2y)cos(x+ 2y)

x2 +y2

lim(x,y)(0,0)

cos(x+ 2y) = 1;


49/236

| sin z| |z| z R

x2 x2 +y2

0

lim(x,y)(0,0)

2| sin(x2y)|x2 +y2

lim(x,y)(0,0)

2x2|y|x2 +y2

lim(x,y)(0,0)

2|y|= 0.

f0 |f| 0

lim(x,y)(0,0)

2 sin(x2y)

x2 +y2 = 0

lim(x,y)(0,0)

2 sin(x2y)cos(x+ 2y)x2 +y2

= 0.

lim(x,y)(0,0)

xyex sin(/4 +xy)

2x2 +y2 .

f(x, y) =xyex sin(/4 +xy)

2x2 +y2

R

2 \ {(0, 0)}

x

y

y = x

f(x, x) =x2ex sin(/4 +x2)

3x2 =

ex sin(4

+x2)

3 1

2/2

3 =

2

6 ,

lim(x,y)(0,0)

f(x, y)

lim(x,y)(1,0)

y2 log x

(x 1)2 +y2


50/236

f0 |f| 0

y

2

z2 +y2 1 (y, z) R2

0 lim(x,y)(1,0)

y2| log x|(x 1)2 +y2 = lim(z,y)(0,0)

y2| log(z+ 1)|z2 +y2

lim(z,y)(0,0)

| log(z+ 1)|= 0.

lim(x,y)(1,0)

y2 log x

(x 1)2 +y2 = 0.

x= 1 + cos , y = sin .

lim0

2 sin2 log(1 + cos )

2 = lim

0 sin2 cos = 0

sin2 cos

f

(x, y)(0, 0) :

1) f(x, y) =xey/x

2) f(x, y) =x3 +y3

x2 +y4

y= x

lim(x,y)0

x ey/x = limx0

x e1 = 0.

y=x,

lim(x,y)0

x ey/x = limx0

x e1x = +.

f

y= x

lim(x,y)0

x3 +y3

x2 +y4= lim

x02x3

x2 +x4= lim

x02x

1 +x2 = 0.


51/236

y=

x,

lim(x,y)0

x3 +y3

x2 +y4= lim

x0x3 +x

x

2x2 = lim

x0x

x+ 1

2

x = +.

f(x, y) =

y2

x x= 0

0 x= 0

f

(0, 0).

D:={(x, y) R2 :|y| x1}

x

= 0

x= 0.

x0 y

f(x, y)

y.

y=

x

f(x, y)

x

D

0limx0

y2

xlim

x0x= 0,

x D x 0.

f

D.

f(x, y) = 2x y3x+ 4y

limx0

limy0

f(x, y)

= lim

y0

limx0

f(x, y)

limx0

limy0

f(x, y)

= lim

x0

2

3

=

2

3


52/236

limy0

limx0

f(x, y)

= lim

y0

1

4

=1

4.

1) lim(x,y)(0,0)

1 exy2x4 +y4

2) lim(x,y)(0,0)

1 cos(xy)log(1 +x2 +y2)

limz0

ez 1z

= 1.

lim(x,y)(0,0)

1 exy2xy2

=1.

(x, y) R2 x4 +y4 y4,

0 lim(x,y)(0,0) |x| y

2x4 +y4

lim(x,y)(0,0) |x| y

2y4

= lim(x,y)(0,0) |x|= 0.

f0 |f| 0

lim(x,y)(0,0)

xy2x4 +y4

= 0;

lim(x,y)(0,0)

1 exy2

x4 +y4= lim

(x,y)(0,0)1 exy

2

xy2 xy2

x4 +y4= 0.

lim(x,y)(0,0)


= 0.

limz0

1 cos zz2

=1

2; lim

z0log(1 +z)

z = 1.


53/236

lim(x,y)(0,0)

1 cos(xy)x2y2

=1

2

lim(x,y)(0,0)

x2 +y2

log(1 +x2 +y2)

= 1.

y2

x2 +y2 1,

0 lim(x,y)(0,0)

x2y2

x2 +y2 lim

(x,y)(0,0)x2 = 0.

lim(x,y)

(0,0)

x2y2

x2

+y2

= 0.

lim(x,y)(0,0)


= lim(x,y)(0,0)

1 cos(xy)x2y2

x2y2

x2 +y2 x

2 +y2

log(1 +x2 +y2)= 0.

f(x, y) = x2 (y x)(x

2

+y2

)

, (x, y)

= (0, 0).

lim(x,y)(0,0)

f(x, y)

= 1

= 2.

= 1.

f(x, y) =

x2 (y

x)

x2 +y2 .

lim(x,y)(0,0)

x2 y

x2 +y2 lim

(x,y)(0,0)x3

x2 +y2

x2 x2 +y2.


54/236

0 lim(x,y)(0,0)

x2 |y|x2 +y2

lim(x,y)(0,0)

|y|= 0.

|f| 0f0

lim(x,y)(0,0)

x2 y

x2 +y2 = 0.

0 lim(x,y)(0,0)

|x3|x2 +y2

= lim(x,y)(0,0)

|x| x2x2 +y2

lim(x,y)(0,0)

|x|= 0

lim(x,y)(0,0)

x3

x2 +y2 = 0.

lim(x,y)(0,0)

f(x, y) = 0

= 2.

y= 2 x.

f(x, 2 x) =

x2 (2 x

x)

(x2 + 4 x2)2 =

x3

25 x4 =

1

25 x

limx0+

f(x, 2 x) = + limx0

f(x, 2 x) =.

f(x, y) :=x1

(x3

+y2

)cos(1 tan x)

(x, y)(0, 0)

tan x

x=

2 +k

k Z

f

f= (x, y) R

2 :k 2

< x arctan 1 +k , k Z x= 0 .


55/236

f

(x, y)(0, 0),

lim(x,y)(0,0)

(x3 +y2) cos

1 tan xx

limx0

f(x,

x) = limx0

(x3 +x) cos

1 tan xx

= limx0

(x2 + 1) cos

1 tan x= cos 1.

limx0

f(x, x) = limx0

(x3 +x2) cos

1 tan xx

= limx0

(x2 +x) cos

1 tan x= 0.

f(x, y) :=(x3 +y2) sin(

1 tan x)

x2 +y2

(x, y)(0, 0)

tan x x= 2 +k k Z

f

f=

(x, y)R2 : k

2 < xarctan 1 + k k Z (x, y)= (0, 0)

.

f

(x, y)(0, 0),

lim(x,y)(0,0)

(x3 +y2) sin

1 tan xx2 +y2

limx0

f(x,

x) = limx0

(x3 +x) sin

1 tan xx2 +x

= limx0

x2 + 1

x+ 1

sin

1 tan x= sin 1.

limx0

f(x, x) = limx0

(x3 +x2) sin

1 tan x2x2

= limx0

x+ 1

2

sin

1 tan x= 1

2 sin1.


56/236

f(x, y) :=x1[(sin x)2 +y2]tan(ex+y)

(x, y)(0, 0)

tan ex+y

ex+y =

2+k

k Z.

k Z

ex+y = 2

+k

k Z+ {0}.

f

f=

(x, y) R2 :x+y= log

2

+k

k Z+ {0} x= 0

.

f

(x, y)(0, 0),

lim(x,y)(0,0)

((sin x)2 +y2) tan(ex+y)

x

limx0

f(x,

x) = limx0

[(sin x)2 +x] tan(ex+x)

x = tan 1.

limx0

f(x, x) = limx0

((sin x)2 +x2) tan e2x

x = 0.

f(x, y) :={[(sin x)3 +y2] tan(ey)}/x

(x, y)(0, 0)

tan ey

ey =

2+k

k Z. k Z

ey = 2

+k , k Z+ {0}.


57/236

f

f=

(x, y) R2 : x= 0 y= log

2

+k

, k Z+ {0}

.

f

(x, y)

(0, 0),

lim(x,y)(0,0)

((sin x)3 +y2) tan(ey)x

limx0

f(x,

x) = limx0

((sin x)3 +x) tan(e

x)

x = tan 1.

limx0

f(x, x) = limx0

((sin x)3 +x2) tan(ex)

x = 0.

z =f(x, y) = sin(x + y2)

(0,

, 0)

x= y = 2z.

fx(x, y) = cos(x+y2), fy(x, y) = 2y cos(x+y

2),

fx(0,

) =

, fy(0,

) =

2

.

(0, , 0)

z=x 2(y ).

x= y = 2z

(0,

, 0)

x= 2t

y =

+ 2t

z=t t R.


58/236

x,y,z

t

t=2t 4t, t R,

=12 2.

= (fx(0,

), fy(0,

), 1) = (, 2

, 1);

= (2, 2, 1)

, = 0

2+ 4+ 1 = 0, =12 2.

.

f(x, y) = log(1 +

x2 + 2y); f

(2, 0)

f

(x, y).

f(x, y) =

f

x(x, y),

f

y(x, y)

=

1

1 +

x2 + 2y

2x

2

x2 + 2y,

1

1 +

x2 + 2y

2

2

x2 + 2y

f(x, y) =

xx2 + 2y+x2 + 2y

, 1

x2 + 2y+x2 + 2y

f(2, 0) = 23 ,13 .

f

(x0, y0)

z= f(x0, y0) +f

x(x0, y0)(x x0) + f

y(x0, y0)(y y0).

x0 = 2, y0= 0, f(2, 0) = log 3

f

(2, 0)

z= log 3 +2

3(x 2) +1

3y


59/236

2x+y 3z= 4 3log3.

f(x, y) = xy

x+y

f

x=

y(x+y) xy(x+y)2

= y2

(x+y)2;

f

y =

x(x+y) xy(x+y)2

= x2

(x+y)2.

f(x, y) = (x+y2)log(x y)

f

x= log(x y) + x+y

2

x y ; f

y = 2y log(x y) x +y

2

x y.

f(x, y) =

xy

x2 +y2 (x, y)= (0, 0)

0 (x, y) = (0, 0)

(x, y)= (0, 0)

fx = y(y

2

x2

)(x2 +y2)2; fy =x(y2

x2

)(x2 +y2)2 .

(x, y) = (0, 0)

f

x(0, 0) = lim

h0f(h, 0) f(0, 0)

h = 0;

f

y(0, 0) = lim

h0f(0, h) f(0, 0)

h = 0.


60/236

f(x, y) =

arctan

x

y y= 0

2 y = 0

y= 0

f

x=

1

1 + x2

y2

1y

= y

y2 +x2;

f

y =

xx2 +y2

.

y = 0

f

x(x, 0) = lim

h0f(x+h, 0) f(x, 0)

h = 0;

f

y(x, 0) = lim

h0f(x, h) f(x, 0)

h = lim

h0arctan x

h

2

h .

x = 0

h

0

x >0

h0 x


61/236

f

f(x, y) = sin x cos y P = (/3, /3)

f(x,y,z) = log(xy/z) P = (3, 2, 2)

f(x, y) = log(2x2 3y2)

f(x, y) =xexy.

f

z=f(x/y)

x z

x+ y

z

y= 0.

z= sin(xy)

x= /3

y =1

z=x2 4xy 2y2 + 12x 12y 1


62/236

f(x, y) =

x+1

2x2y y0

exy 1y

y 0

f

y(x, 0) = lim

h0f(x, h) f(x, 0)

h = lim

h0x+ 1

2x2h xh

=x2

2

h 0

f

x(x, y) = 1 +xy


63/236

y0,

y 0

f

y(x, y) =

x2

2

y


64/236

lim(h,k)(0,0)

h2k3

(h4 +k4)

h2 +k2

h= k

lim(h,k)(0,0)

h2k3

(h4 +k4)

h2 +k2= lim

h0h5

2

2h4|h|

(0, 0).

f(x, y) = arctan

x

y y= 0

2 y = 0

(x, y)

y= 0.

(x, 0)

xR.

f

x(x, 0) = lim

h0f(x+h, 0) f(x, 0)

h = 0

f

y (x, 0) = limh0f(x, h)

f(x, 0)

h = limh0arctan x

h

2h

x

h 0+

h0.

(x, 0)

x R.

f(x, y) =|x| log(1 +y)

f(0, 0) = 0.

f

x(0, 0) = lim

h0f(h, 0) f(0, 0)

h = 0,

f

y(0, 0) = lim

h0f(0, h) f(0, 0)

h = 0.

lim(h,k)(0,0)

f(h, k)h2 +k2

= lim(h,k)(0,0)

|x| log(1 +y)h2 +k2

lim(h,k)(0,0)

|x||y|h2 +k2

lim(h,k)(0,0)

1

2

h2 +k2;


65/236

f

(0, 0).

f(x, y) =xy

d2f(1, 2)

fx(x, y) =yxy1, fy(x, y) =x

y log x,

fxx(x, y) =y(y 1)xy2, fxy(x, y) =xy1(1 +y log x), fyy(x, y) =xy log2 x

fxx(1, 2) = 2, fxy(1, 2) = 1, fyy(1, 2) = 0

d2f(1, 2) = 2dx2 +dx dy.

f(x, y) = x

1 +y y1 +x

fx(x, y) = 1

1 +y y

2

1 +xfy(x, y) =x

2(1 +y)3/2 1 +x

fxx(x, y) =y4

(1 +x)3

2 fxy(x, y) =12

(1 +y)

3

2 + (1 +x)3

2

fyy(x, y) =

3

4(1 +y)

5

2 ;

fx(0, 0) = 1 fy(0, 0) =1

fxx(0, 0) = 0 fxy(0, 0) =1 fyy(0, 0) = 0


66/236

df(0, 0) =dx dy; d2f(0, 0) =2dxdy.

y

0

11 +y

= (1 +y)1/2 = 1 12

y+o(y)

x0

1 +x= 1 +

1

2x+o(x)

f(x, y) =x y xy+xo(y) +yo(x).

|xy|x2 +y2

12

(x, y) R2

|xo(y)|x2 +y2

12

o(y)y 0

(x, y)(0, 0) xo(y) x2 + y2.

yo(x).

f(x, y) =x y xy+xo(y) +yo(x)

df(0, 0) =dx dy; d2f(0, 0) =2dxdy.

xy

1 +x2 +y2


67/236

f(x, y) = e

1

1x2y2 x2 +y2


68/236

fx(x, y) = 3y4e3x, fy(x, y) = 4y

3e3x

fx(0, 1) = 3, fy(0, 1) =4

f(0, 1) = (3, 4), ||f(0, 1)||= 5.

3

5, 4

5

.

1

=4

5 ,

3

5

,

2

=

4

5 , 3

5

.

= (cos , sin )

[0, 2)

D

f(0, 1) . f

D

f(0, 1) =D() = 3 cos 4sin .

D() = 0

(cos , sin ) =

45

,35

D() = 0

(cos , sin ) =

3

5, 4

5

.

3

5, 4

5

3

5,4

5

f(x, y) = 3

x2(y 1) + 1

Dvf(0, 1) v R

2


69/236


70/236


71/236

f(x, y) =ex2

(x

y3)

R;

(0, 1)

y = (x+ 1)2

x;

(0, 1)

x

2 =

y

3=z.

f(x, y) =y4 + 2xy3 +x2y2

(0, 1)

a)i + 2j b)j 2i c)3i d)i +j.

f(a, b)

f(x, y)

D(i+j)/

2f(a, b) = 3

2 D(3i4j)/5f(a, b) = 5.

(x, y)

xy T

T(x, y) =x2 ey.

(2, 1)

T


72/236

f(x, y) =

x3y

x2 +y2 (x, y)

= (0, 0)

0 (x, y) = (0, 0)

fxy(0, 0) = 1=fyx(0, 0) = 0. f C2?

f C2(R2)

fx(x, y) =x sin y fy(x, y) =y cos x

f(x, y) =

xy

x2 y2x2 +y2

(x, y)= (0, 0)0 (x, y) = (0, 0).

f C1(R2)


73/236


74/236

f(x, y) =

|y

|sin(x2 +y2)

R2

f

R2

y >0

f

x= 2xy cos(x2 +y2)

f

y= sin(x2 +y2) + 2y2 cos(x2 +y2).

y 0

sin x2

h


75/236

f(x, y) = |

y

| cos x

x2

+y2

(x, y)

= (0, 0)

0 (x, y) = (0, 0)

f(x,y,z) =

z4(x2 +y2)x2 +y2 +z2

(x,y,z)= (0, 0, 0)0 (x,y,z) = (0, 0, 0)

lim(x,y)(0,0)

|y| cos xx2 +y2

.

lim(x,y)(0,0)

|y| cos xx2 +y2

= lim0

1| sin | cos( cos );

> 1

1

y 0

1/2 = 1 + 1.

= (cos , sin )

limt0

f(t cos , t sin ) f(0, 0)t

= limt0

|t|1| sin | cos(t cos )t

=

0< 2

0< 2.

2

f

>2,

lim(h,k)(0,0)

f(h, k) f(0, 0) hfx(0, 0) kfy(0, 0)h2 +k2

= lim0

2| sin | cos( cos ) = 0.

f

>2.


76/236

f(x, 0, 0) = 0 x R, f(0, y, 0) = 0 y R, f(0, 0, z) = 0 z R,

fx(0, 0, 0) =fy(0, 0, 0) =fz(0, 0, 0) = 0

f(0, 0, 0) = (0, 0, 0).

= (h1, h2, h3)

lim

(0,0,0)f(

) f(0, 0, 0) f(0, 0, 0),

|| || = lim (0,0,0)

h43(h21+h

22)

(h21+h

22+h

23)

3.

R

3 :

h1= sin cos , h2= sin sin , h3= cos

lim

(0,0,0)f(

) f(0, 0, 0) f(0, 0, 0),

|| || = lim0

2+1(cos )4(sin )2 = 0 >0

2+1 0 |(cos )4(sin )2| 1. f (0, 0, 0)

> 0;

f

(0, 0, 0).

f(x, y) =

x2y3

x4 +y4 (x, y)= (0, 0)

0 (x, y) = (0, 0)

(x, y) R2

(x, y)= (0, 0).

0 lim(x,y)(0,0)

x2y3x4 +y4 = lim(x,y)(0,0) |y|(x2

y

2

)x4 +y4 lim(x,y)(0,0) 12 |y|= 0;

lim(x,y)(0,0)

x2y3

x4 +y4 = 0

(x, y)= (0, 0)

f

x(x, y) =

2xy3(x4 +y4) 4x3x2y3(x4 +y4)2

=2xy7 2x5y3

(x4 +y4)2


77/236

f

y(x, y) =

3y2x2(x4 +y4) 4y3x2y3(x4 +y4)2

=3x6y2 y6x2

(x4 +y4)2

f

x(0, 0) =

f

y(0, 0) = 0.

lim(h,k)(0,0)

f(h, k) f(0, 0)h2 +k2

= lim(h,k)(0,0)

h2k3

(h4 +k4)

h2 +k2

h = k :

limh0

h5

2

2h4|h| = limh01

2

2 h|h|

h >0

1

2

2

h


78/236

lim0

[cos3 + 3 | cos | cos sin + 2 sin4 ] = 0.

f

(0, 0).

f

x(0, 0) = lim

h0f(h, 0) f(0, 0)

h = lim

h0h3

h21

h= 1

f

y(0, 0) = lim

h0f(0, h) f(0, 0)

h = lim

h0h4

h21

h= 0.

= (v1, v2) =

2

2 ,

2

2 ,

D

f(x, y) = limt0

f(t v1, t v2) f(0, 0)t

= limt0

t3 12

2+ 3 t2 |t| 1

2

2+ t4 1

4

t21

t

= limt0

1

2

2+ 3

|t|t

1

2

2+

t

4

2

t0+ 1

2

t0.

f(x, y) =

|x| sin y x= 00 x= 0

f

f

f

x= 0;

.

0

0 lim(x,y)(0,0)

|x|| sin y| lim(x,y)(0,0)

|x||y|= 0

f


79/236


80/236

R2

f(x, y) =

x3 + 3xy2 +y4

x2 +y2

(x, y)= (0, 0)0

.

f

(0, 0);

f

(0, 0)

f

(0, 0).

lim(x,y)(0,0)

x3 + 3 x y2 +y4

x2 +y2 = lim

03 cos3 + 3 3 cos sin2 + 4 sin4

2

= lim0

[ cos3 + 3 cos sin2 + 2 sin4 ].

0

lim0 |

cos3

|= lim

0

|cos3

| lim0

= 0

0lim0

|3 cos sin2 |= lim0

3 | cos sin2 | lim0

3 = 0

0lim0

|2 sin4 |= lim0

2 sin4 lim0

2 = 0.

f0 |f| 0

lim0

cos3 = 0 lim0

3 cos sin2 = 0 lim0

2 sin4 = 0

lim0

[cos3 + 3 cos sin2 + 2 sin4 ] = 0.

f

(0, 0).

f

x(0, 0) = lim

h0f(h, 0) f(0, 0)

h = lim

h0h3

h21

h= 1

f

y(0, 0) = lim

h0f(0, h) f(0, 0)

h = lim

h0h4

h21

h= 0.


81/236

f

(0, 0)

lim(h,k)

(0,0)

f(h, k) f(0, 0) h fx(0, 0) k fy(0, 0)h2 +k2

.

lim(h,k)(0,0)

f(h, k) f(0, 0) h fx(0, 0) k fy(0, 0)h2 +k2

= lim(h,k)(0,0)

h3+3hk2+k4

h2+k2 0 h 0h2 +k2

= lim(h,k)(0,0)

h3 + 3hk2 +k4 h3 hk2(h2 +k2)

h2 +k2

= lim(h,k)(0,0)

2hk2 +k4

(h2 +k2)

h2 +k2.

g(h, k) = 2hk2 +k4

(h2 +k2)

h2 +k2

h= k.

g(h, h) = 2h3 +h4

2h22 |h|=

2h+h2

2 2 |h|.

h0+ g(h, h)1/2, h0 g(h, h) 1/2

f

f(x, y) =

xy

x2 +y2 (x, y)= (0, 0)

0 .

(0, 0)


82/236

f(x, y) =

3yey2

/x4

x= 00 x= 0

f

f

Dvf(0, 0) = v Rn ||v||= 1

f

f(x,y,z) =

(x2 +y2 +z2)sin(x2 +y2 +z2)1/2 (x,y,z)= (0, 0, 0)0 (x,y,z) = (0, 0, 0)

f(x, y) =

x2 +y2 x= 0y x= 0

(0, 0, 0)


83/236

2x3 +y4 +z3 xz 2x= 0

z = g(x, y)

(1, 0, 0).

(1, 0, 0)

z g(x, y) = 0.

P= (1, 0, 0).

f(x,y,z) := 2x3 + y4 + z3 xz 2x. f C(R3), f(1, 0, 0) = 0 fz(x,y,z) = 3z2 x fz(1, 0, 0) =1= 0.

I

(1, 0)

g:I R

f(x,y,g(x, y)) = 0

(x, y)I .

g(1, 0) = 0

g

x(1, 0) =

f

x(1, 0, 0)

f

z(1, 0, 0)

= 4 g

y(1, 0) =

f

y(1, 0, 0)

f

z(1, 0, 0)

= 0.


84/236

z=g(1, 0) +g

x(1, 0) (x 1) + g

y(1, 0) (y 0)

z= 4(x 1).

y=h(x)

x cos y=y2

h(x)

x, h(x)

f(x, y) :=x cos y y2.

y=h(x)

x

f C

1

f(x, y) = 0

fy(x, y) =x sin y 2 y= 0

x cos y y2 = 0xsin y 2 y= 0.

h(x)

cos y = 0

y = 0

cos y= 0

x

y

x=

y2

cos y

y [y tan y 2] = 0.

y = 0

y tan y = 2.

(0, 0);

f(x, y) = 0

h(x).

y

y tan y = 2

j(y) :=

y tan y 2. y0,

y=/2 +k

k Z.

j(y) = tan y+ y

cos2 y =

sin y cos y+y

cos2 y .

x/2 sin y cos y0 y > /2 j(y)> /2 1

j(y)> 0

y,

j


85/236

j(y) =[1 + cos2 y sin2 y] cos2 y+ 2 cos y sin y (x+ sin y cos y)

cos4 y

=cos2 y+ cos4 y sin2 y cos2 y+ 2 x cos y sin y+ 2 cos2 y sin2 y

cos4 y

=2 y cos y sin y+ cos2 y sin2 y+ cos4 y+ cos2 y

cos4 y

=2 y cos y sin y+ cos2 y [sin2 y+ cos2 y] + cos2 y

cos4 y

=cos y [2 cos y+ 2 y sin y]

cos4 y = 0cos y+y sin y= 0

j.

y;

f(x, y) = 0

h(x).

(x, y)

h(x) =fx(x, h(x))fy(x, h(x))

= cos h(x)xsin h(x) 2 h(x) = cos h(x)

xsin h(x) + 2 h(x).


86/236


87/236

F(x, y) =F(x, y) 1 =x ey y ex 1.

G

F(x, G(x)) = 0

F

F.

G(x) =

Fx(x, G(x))

Fy(x, G(x))

Fx(x, y) =ey y ex Fx(x, G(x)) =eG(x) G(x) ex

Fy(x, y) =x ey ex Fy(x, G(x)) =x eG(x) ex.

G(x) =eG(x) G(x) exx eG(x) ex .

d

du(G(H(u))) =G(H(u)) H(u)

H(u) = 2 sin ucos u

d

duG(H(u)) =e

G(H(u)) G(H(u)) eH(u)H(u) eG(H(u)) eH(u) H

(u) =eG(sin2 u) G(sin2 u) esin2 usin2 u eG(sin

2 u) esin2 u (sin(2u)).

F(x, y) = 0.

A={(x, y)R2 : F(x, y) = 0}. F C1(A) F C1(R2) Fy(x, y)= 0 (x, y) A

x ey y ex 1 = 0x ey ex = 0

ex(1 y) 1 = 0.

(x, y)R2

y 1,

x

(x, y)

y 1

1 y0 ex(1 y)0 ex(1 y) 1< 0.

y < 1.

x

y

x= log

1

1 y


88/236


89/236

y

x

x0= 0

: IJ

(0) =f

x (0, 0)fy

(0, 0)= 0.

(x) =fx

(x, (x))fy

(x, (x))

(x) =

2fx2

(x, (x)) + 2f

xy(x, (x))(x)

fy

(x, (x)) fx

(x, (x))

2fxy

(x, (x)) + 2f

x2(x, (x))

fy (x, (x))2 .

(0) = 2f

x2(0, 0)

f

y(0, 0).

2f

x2(x, y) = 1 2 (1 +x

2 +y2) 4x2(1 +x2 +y2)2

, 2f

x2(0, 0) =1,

(0) =(1) = 1> 0 0

f(x, y) = 0

y = (x)

x0 (x0) =y0

(x0)

f(x, y) =x+ 2y+x sin y, (x0, y0) = (0, 0)

f(x, y) =xey +y+ 2, (x0, y0) = (0, 2)

f(x, y) =xy + log(xy) 1, (x0, y0) = (1, 1)

f(x, y) =y5 + logx+y

2

xy, (x0, y0) = (1, 1)

f(x, y) =x2y22x+2y= 0


90/236

y= y(x)

x2y2 +y3 +x+y = 0,

limx0

y(x) +x

x2 .

y logxy +y2 x= 0

y= y(x)

(1, 1).

limx1

y(x) 1(x 1)2 .

x3 +y3

3x+y = 0

y= y(x)

R.

x2 +y ex2y = 0

y =y(x)

R.


91/236

y3 +y2 +x2 +x+y = 0

y = y(x)

R.

y=y(x)

[2, 1].

limx0

y(x) +x

x2 .

2x+ yx

e(tx)2 dt= 0

y= y(x)

x= 0.

limx0

y(x) +x+x2

cos x 1 .

x cos(xy) = 0

y= y(x)

1,

2

.

limx0

2y(x) (2 x)(x 1)2 .

ex+y+cos(x+y) +ex+y+sin(x+y) e= 1

y= y(x)

R


92/236

xy2 +z3

xy+ 2z= 0

z= g(x, y)

R2.

y= y(x)

x2 +x(y2 1) +y(y2 + 1) = 0

limx1

y(x) +x 1(x 1)2 .

x2 +x(y2 1) +y(y2 + 1) = 0

y= y(x)

x= 0,

limx0

y(x) xx2

.

x3 +y3 +x2 xy+x+y = 0

y= y(x)

x= 0,

limx0

y(x) +x

x2 .


93/236

(x+y)3

x+y = 0

y= y(x)

R.

x2 y ex2+y = 0

y =y(x)

R.

x sin(xy) = 0

y= y(x)

(1, 0).

limx1y(x)

(x 1)2 .

(x+y)3 3(x y) + 2 = 0

y =y(x)

R.

y = h(x)

x sin y=y2

h(x)

x, h(x)


94/236


95/236

0x2+2y

et2

dt= 0

y= y(x)

R.

limx1

y(x) +x2 x+ 12

(x 1)2 .

x4

+ 2y3

+z3

yz 2y= 0

z = g(x, y)

(0, 1, 0)

(0, 1, 0)

z g(x, y) = 0

z= 4(y 1).

x4 + 2y3 +z3 yz 2y= 1

z = g(x, y)

(0, 0, 1)

(0, 0, 1)

z g(x, y) = 0

z= 1 +1

3y.

2x2 + 2y3 +z3 yx 2y= 1

z = g(x, y)

(0, 0, 1)

(0, 0, 1)

z g(x, y) = 0

z= 1 +2

3y.


96/236

(x 1) log(cos y) + (y 1) e(x2

) = 0

y

x,

y = y(x),

(1, 1).

y(1).

y(1) = log(cos 1)e1

.

(x 1) sin(sin y) + (y 1) sin(x2) = 0

y

x,

y = y(x),

(1, 1).

y(1).

y(1) =

sin(sin 1)

sin1

.

f(u, v) := uev veu 1

(u, v) R2,

g

f(u, g(u)) = 0

u R.

g(u)

h(z) :=

1

(z3 + 1)

z R, ddz

[g(h(z))].

d

dzg(h(z)) =e

g(h(z)) g(h(z)) eh(z)h(z) eg(h(z)) eh(z) h

(z) =eg

1

z3+1

g 1

z3+1

e

1

z3+1

1z3+1

eg

1

z3+1

e 1z3+1

3 z

2

(z3 + 1)2

.


97/236

k

(a)q1(x, y) =k2x2 + (k+ 1)y2 + 12xy,

(b)q2(x,y,z) =x2 +y2 + 2z2 + 2kxz+ 2yz

k = 0 a = k2 > 0k det A =(k3)(k2 + 4k + 12)

k > 3

det A > 0

k < 3

det A < 0

k >3

k 0

k = 0

(0, 6), (6, 1)

c = 1 > 0

det A=36


98/236

f(x, y) =x4 +y4

2(x

y)2 + 2.

C(R2),

f

f.

f

x(x, y) = 4x3 2(x y)2 = 4x3 4x+ 4y f

y(x, y) = 4y3 + 4(x y)

4x3

4(x y) = 04y3 + 4(x y) = 0.

x3 +y 3 = 0

x =y.

4y3 8y= 0 y[y2 2] = 0 y= 0 y =

2.

(0, 0) (2, 2) (2, 2).

2f

x2(x, y) = 12x2 4

2f

xy(x, y) =

2f

yx(x, y) = 4

2f

y 2(x, y) = 12y2 4.

Hf(

2, 2) =Hf(

2,

2) =

24 4 44 24 4

= 20 44 20

Hf(

2, 2) =

Hf(

2,

2) = 400 16> 0

fxx>0.

(

2, 2) (2, 2)

Hf(0, 0) =

4 4

4 4


99/236


100/236

2t

x2

(x, y) = 12x2

8

2t

xy

(x, y) = 2t

yx

(x, y) = 0 2t

y2

(x, y) = 6y

6.

Ht(0, 0) =

8 0

0 6

Ht(0, 0) = 48 > 0 txx 0

txx(

2, 2) = 16 > 0

(2, 2)

f(x, y) = sin(x+y)

cos(x

y).

f(x, y) = sin(x+y) cos(x+y) = sin x cos y+ cos x sin y cos x cos y sin x sin y

=sin x[cos y sin y] cos x[cos y sin y] = [cos y sin y] [sin x cos x]

=2

2

2 (cos y sin y)

2

2 (sin x cos x)

= 2 sin

x

4

cos

y+

4


101/236

1sin

x 4

1 1cos

y+

4

1

f

x=3

4+ 2k k Z

y=4

+ 2h h Z

x=7

4+ 2k k Z

y=3

4+ 2h h Z.

f

x=

3

4+ 2k k Z

y =3

4+ 2h h Z

x= 74

+ 2k k Zy=

4+ 2h h Z.

f

f

fx(x, y) = 2 cos

x 4

cos

y+

4

= 0

fy(x, y) =2sin

x 4

sin

y+

4

= 0.

x=3

4+k k Z

y =4

+h h Z

x=

4+k k Z

y=3

4+h h Z.


102/236

f

f(x, y) = (x2 +y2)e(x2+y2).

f

r =

x2 +y2.

f

g(r) =r2

er2

r0.

g(0) = 0

g(r) 0

r

g(r) > 0

r= 0.

limr+

g(r) = 0

r= 0

g(r) = 2r(1 r2)er2

g(r) = 0

r= 1.

r= 1

g

f

r = 0

(0, 0)

r = 1

(0, 0).

(0, 0)

f(x, y) =x 3

(y x)2.

f

R2.

{(x, y) R2 :y = x} f f

x(x, x) = lim

h0f(x+h, x) f(x, x)

h = lim

h0(x+h) 3

h2

h = lim

h0xh1/3 + lim

h03

h2.

x= 0

f

(x, x)

x= 0.

f

x(0, 0) = lim

h0f(h, 0) f(0, 0)

h = lim

h0h 3

(h)2h

= 0


103/236

f

y(0, 0) = lim

h0f(0, h) f(0, 0)

h = 0

lim(h,k)(0,0)

f(h, k) f(0, 0) hfx(0, 0) kfy(0, 0)h2 +k2

= lim(h,k)(0,0)

h 3

(k h)2h2 +k2

(0, 0) f

fx(x, y) = 3y 5x3

y x fy(x, y) = 2x

3

y x

f

R2 \ {(x, y) : y = x}.

y = x

f(x, y) = 0.

f(x, y)

f(x, y)0

x >0

f(x, y)0

x 1}.

f

C(R2).

fx(x, y) = 2x(log(1 +y) +y2) = 0

fy(x, y) =x2

1

1 +y+ 2y

= 0.


104/236

x= 0;

log(1 +y) +y2 = 0

1

1 +y+ 2y= 0.

y = 0

2y2 + 2y+ 1 = 0

(0, k)

k >1. f(x, y) = 0.

fxx(x, y) = 2(log(1 +y) +y2) fyy(x, y) =x

2

2 1

(1 +y)2

fxy(x, y) =fyx(x, y) = 2x

1

1 +y+ 2y

;

Hf(0, k) =

log(1 +k) +k2 0

0 0

f(x, y) =f(x, y) f(0, k) =f(x, y) =x2 log(1 +y) +x2y2 =x2[log(1 +y) +y2].

log(1 +y) +y2 =:g(y).

g(y)< 0 1< y 0y >0.

k >0

(0, k)

k= 0

1< k 0 1< y0


105/236

f(x, y) =x2(y

1)3(z+ 2)2.

C(R3).

fx(x,y,z) = 2x(y 1)3(z+ 2)2 = 0fy(x,y,z) = 3x

2(y 1)2(z+ 2)2 = 0fz(x,y,z) = 2x

2(y 1)3(z+ 2) = 0.

x= 0, y = 1

z=2,

(0, h , k), (l, 1, k), (l,h, 2) l ,h,k R. f

f

(l, 1, k)

y = 1

f

(0, h , k) (l,h, 2) h < 1 f

h 1

f

h >1.

f(x,y,z) = 1

x+1

y +1

z+xyz.

C(R3 \ {(x,y,z) :x = 0, y= 0, z= 0}).

fx(x,y,z) = 1x2

+yz= 0

fy(x,y,z) =1y2

+xz= 0

fz(x,y,z) =1

z2 +xy = 0.

x,

y

1

y 1

x= 0 x y = 0;

y,

z

1

y 1

z = 0 x = y = z.


106/236

1x2

+x2 = 0 x4 = 1.

(1, 1, 1)

(

1,

1,

1).

fxx(x,y,z) = 2

x3 fxy(x,y,z) =fyx(x,y,z) =z fyy(x,y,z) =

2

y3

fzz(x,y,z) = 2

z3 fxz(x,y,z) =fzx(x,y,z) =y fyz(x,y,z) =fzy(x,y,z) =x.

Hf(1, 1, 1) =

2 1 1

1 2 1

1 1 2.

H1= 2> 0 H2= 3> 0 H3= 4> 0

(1, 1, 1)

f.

Hf(1, 1, 1) =

2 1 11 2 11 1 2.

H1 =2< 0 H2= 3> 0 H3=4< 0

(1, 1, 1)

f.

f(x,y,z) =f(x, y, z).

k,

q(x,y,z) =x2 + 2kxy +y2 + 2kyz+z2;

q

A=

1 k 0

k 1 k

0 k 1.


107/236

2kxy

a12=k

a21

A1= 1 A2= 1

k2 A3= A= 1

2k2.

|k|> 12

A3 0

|k|< 12

Ai

|k|= 12

q(x,y,z) = x+ 1

2

y2

+ 1

2

y+z2

q(x,y,z)0

(x,y,z) Rn

q(x,y,z) = 0

1

2h,h, 1

2h

.

f(x, y) =x3y2 x4y2 x3y3.

C(R2)

f

f.

f

x(x, y) = 3x2y2 4x3y2 3x2y3 f

y(x, y) = 2x3y x42y x33y2

x2y2[3 4x 3y] = 0x3y[2

2x

3y] = 0.

(k, 0) (0, h) k, h R.

3 4x 3y= 02 2x 3y= 0.


108/236

12

, 13

.

1

2,1

3

(k, 0) (0, h)

k, hR.

2f

x2(x, y) = 6xy2 12x2y2 6xy3

2f

xy(x, y) =

2t

yx(x, y) = 6x2y 8x3y 9x2y

2f

y 2(x, y) = 2x3 2x4 6x3y.

Hf(k, 0) =

0 0

0 2k3 2k4

Hf(k, 0) = 0

f(k, 0) = 0,

f(x, y) =f(x, y) f(0, 0) =f(x, y) =x3y2 x4y2 x3y3 =x3y2[1 x y].

f

f

(k, 0)

k > 1 k < 0

f

0< k 0

1

2,1

3


109/236

f.

f(x, y) =|xy|(x+y 1).

f

(0, h)

(k, 0)

h, k R

(0, 0),(0, 1)

(1, 0)

f

x(x, y) = 2xy+y2 y

f

y(x, y) =x2 + 2xy x

f

x(x, y) =2xy y2 +y


110/236

f

y(x, y) =x2 2xy+x.

f

x

(0, h) = limt0

f(t, h) f(0, h)t

= limt0

|th|(t+h 1)t

h= 0

h= 1

f

x(0, h) = 0,

x

f

y(k, 0) = lim

t0f(k, t) f(k, 0)

t = lim

t0|kt|(k+t 1)

t

k= 0

k= 1

f

y(k, 0) = 0,

y

(0, h)

(k, 0)

k, h= 0, 1

f.

(0, 0)

lim(h,k)(0,0)

f(h, k) f(0, 0) hfx(0, 0) kfy(0, 0)h2 +k2

= lim(h,k)(0,0)

|hk|(h+k 1)h2 +k2

= 0.

(1, 0)

(0, 1)

f.

f.

f

2xy+y2 y = 0x2 + 2xy x= 0

(0, 0), (0, 1), (1, 0)

1

3,1

3

.

fxx(x, y) = 2y fxy(x.y) = 2x+ 2y 1 fyy(x, y) = 2x

Hf

1

3,1

3

=

2

3

1

31

3

2

3


111/236

fxx

1

3,1

3

>0

Hf

1

3,1

3

=

1

3 > 0

1

3,1

3

f C2(A) A

f,

f,

f

f

f.

f

f

h >1

(0, h)

y >x+ 1 f (0, h)

(0, h)

h 1

(0, k)


112/236

f(x, y) = 4y2

4x2y2

y4.

C(R2).

fx(x, y) =8xy2 = 0fy(x, y) = 8y 8x2y 4y3 = 0

(k, 0)

k R

(0, 2).

fxx(x, y) =8y2 fxy =16xy fyy(x, y) = 8 8x2 12y2

(k, 0)

Hf(k, 0) = 0

f(x, y) =f(x, y) f(k, 0) =f(x, y) =y2[4 4x2 y2].

f

f

|k| > 1

(k, 0)

f |k| < 1 (k, 0)

(1, 0)

(0, 2)

Hf(0,

2) =

16 0

0 16

(0, 2)

f.

(x, y) R2

f(x, y) 40

f(x, y) 4 = 4y2 4x2y2 y4 4 =4x2y2 (y2 2)2 0.

f

lim(x,y)(,1)

f(x, y) =


113/236

f.

infR2

f(x, y) =.

f

f(x, y) =ex4

4x2

y+3y

2

.

f C(R2)

ex

g(x, y) = x4

4x2y+ 3y2.

f

g.

gx(x, y) = 4x3 8xy= 0

gy(x, y) =4x2 + 6y = 0


114/236


115/236

f(x, y) =x3 +y3

3xy

(0, 0)

(1, 1)

R3

q1(x,y,z) =x2 + 2y2 + 2z2 + 2xy+ 2xz,

q2(x,y,z) = 2xz

2xy

y2

2yz

q3(x , y, z, t) =2x2 +ky 2 z2 t2 + 2xz+ 4yt+ 2kzt.

R2

1)f(x, y) = 3x2 2y2

2)f(x, y) = (y 2) exy

3)f(x, y) = (x+ 1) exy

4)f(x, y) =ey (x2 + 1) y5)f(x, y) = 3x2y y3 +x26)f(x, y) = 3y3 x2y x27)f(x, y) =4xy+ 4x8)f(x, y) = 4y+ 4xy

9)f(x, y) = log(1 +y2 xy+ 2x2)10)f(x, y) = log(1 + 4y2

2xy+x2)

11)f(x, y) = 2y2 +x2 y12)f(x, y) = arctan(3x2 +y2)

13)f(x, y) =8x2 2y2 2xy+ 2


116/236

1)(0, 0)

2)(1/2, 0)

3)(0, 1)

4)(0, 0)

5)(0, 0)

, (1/3, 1/3) , (1/3, 1/3) 6)(0, 0)

, (3, 1)

, (3, 1)

7)(0, 1)

8)(1, 0) 9)(0, 0)

10)(0, 0)

11)(0, 4)

12)(0, 0)

13)(0, 0)

n

fn(x, y) = (x2 + 3xy2 + 2y4)n n N \ {0}.

f(x, y) = log(1 +x2) x2 +xy2 +y3 + 2.

f(x,y,z) =x sin x+ log(1 +y2) z z

0

et2

dt= 0

(0, 0, 0)

z = z(x, y)


117/236

x2 +xu2 +y2 +exu

z+y2ez = 0

z = z(x,y,u)

z(0, 0, 0) = 1

(0, 0, 0)

f1(x,y,z) = [sin(x z)]2 +y2 xyz;

f

2(x,y,z) = [sin(x

z)]2 +y2 +y2z;

(0, 0, 0)

(0, 0, 0).

f(x, y) = (x4 +y4)ex2+y2

2 .

f(x, y) = x2 + 2y

x2 +y2 + 1.

R

f(x, y) =

(x2 +y2) log(x2 +y2) (x, y)= (0, 0)0 (x, y) = (0, 0).


118/236

R3 :

x= y = 0

x= 3

z= 2y.

k

f(x, y) = 2 +kx2 + 4xy+ (k 3)y2 + (2x+y)4.

f(x, y) = arctan(x2 +y2) log(1 +x2) log(1 +y2).

(y 1)z+ez + (x2 +x)log y 1 = 0

z = z(x, y)

y = 1.

y= 1

f(x,y,z) =x2 2x+y2 + log(1 +z2).

f(x,y,z) = (x2 +y2)2 xy+z2.


119/236

k,

q(x,y,z) =kx2 +ky2 +kz2 + 2xy+ 2yz;

q(x , y, z, t) =x2 +xy y2 +z2 2xz 2yz+kt2.

k

f(x, y) = 5 +kx2 + 2xy+ 4kxz 6y2 3z2.

(xi, yi) i = 1, . . . , n .

a,b,c

ni=1

(ax2i +bxi+c yi)2.


120/236

x3 + 3x2 + 4xy+y2

y2 x2yx2y2(1 x y)x2 +y2 +z3 2x 3zx3 +xy+y2 +yz+z3

|x2 +y2

4y

|+x

xy

+ yx

x4 +ax2y+y2

x+y 1x2 +y2

(2x+y)ex2y2

x4 x3 +y2xy log(xy2) +x2y

x2 +y2 + 1

x+

1

y(x+ 3y)exy

xy

x2 +y2

x3 + 6xy+y2

x2yex+py

x log(x+y)

x2 +y2 + 2z2 +xyz

sin(x+y)cos(x y)xy2 x2 y2


121/236

E

f(x, y) = 2(x+ 2y) x2

2y2

1.

f

E.

sin2 x+y2 + 2axy

f(x, y) = x3y

x4 +y4

E= R2 \ {(0, 0)}.


122/236

f(x, y) =x2 + 2y2 4x+ 4yf(x, y) =xy x+yf(x, y) =x3 +y3 3xyf(x, y) =x4 +y44xy

f(x, y) =x

y+

8

x y

f(x, y) = cos(x+y)

f(x, y) =x sin yf(x, y) = cos x+ cos y

f(x, y) =x2ye(x2+y2)

f(x, y) = xy

2 +x4 +y4

f(x, y) =xex3+y3

f(x, y) = 1

1 x+y+x2 +y2

f(x, y) = 1 + 1x1 +1

y1

x

+1

y

f(x,y,z) =xyz x2 y2 z2f(x,y,z) =xy +x2z x2 y z2

f(x,y,z) = 4xyz x4 y4 z4

(1, 1, 1).

f(x, y) =

xyex2y4 .


123/236

f(x, y) = 1

1 +x2

+y2

f(x,y,z) =

xyzex2y2z2 .

f(x, y) =x+ 8y+

1

xy

x >0, y >0.

z=g(x, y)

e2zxx2 3e2zy+y2 = 2.

f(x, y) = (y x2)(y 3x2).

f

f

f(x,kx)

x = 0

k

f(0, y)

y = 0.

f(x, y)

f : R2 R

f(x, y) = (|x| +y)exy.


124/236

f(x, y) = (x

4

+y

4

)exp

1

2 (x

2

+y

2

)

.

f(x,y,z) =x3 +y3 + 5xy z2 + 2z

f.

f(x, y) =exy +ex +ey+1

g(x, y) =f(x2, y).

D={(x,y,z) : (x 1)2 + (y 2)2 + (z+ 1)2 = 4} f(x,y,z)

f(x,y,z) = (1, 1, 1) (x,y,z)

f

D f

D

f

D f

D

g(x,y,z) = (x 1)2 + (y 2)2 + (z+ 1)2 4 = 0.


125/236

L(x , y, z, ) =f(x,y,z) + g(x,y,z).

L.

L(x , y, z, ) = (0, 0, 0, 0)

1 + 2(x 1) = 01 + 2 (y 2) = 01 + 2(z+ 1) = 0

(x 1)2 + (y 2)2 + (z+ 1)2 = 0

L(x , y, z, ) = (0, 0, 0, 0)

= 12(1x) = 0

1 + (y2)(1x) = 0

1 + (z+1)(1x) = 0

3(x 1)2 = 4

1 2

3, 2 2

3, 1 2

3 1 + 2

3, 2 +

23

, 1 + 23

(0, 0)

g(x, y) = (x+ 5y)(2x y)

10 5 0

f

(0, 0)

fx(x, y) = 4x y+ 10y fy(x, y) =x+ 10x 10y

fxx(x, y) = 4 fxy(x, y) =fyx(x, y) = 10 fyy(x, y) =10


126/236

fxx(0, 0) = 4 fxy(x, y) =fyx(0, 0) = 10 fyy(0, 0) =10

fxx(0, 0) > 0

(0, 0)

f

>0

Hf(0, 0) =40 (10 )2 =2 20 100 =( 10)2 0

f

(0, 0, 0)

F(x,y,z) = 2x2 + y2 + (

1) z2?

1

> 1

> 0

< 1

< 0

0 < < 1

= 0

4, 0, 2

(0, 0, 0)

= 0

= 1

4, 2, 0

(0, 0, 0)

f(x, y) = (y 1)(y x2) R2

fx(x, y) =2xy+ 2x fy = 2y x2 1


127/236

fxx(x, y) =2y+ 2 fxy(x, y) =fyx(x, y) =2x fyy(x, y) = 2

f(x, y) = (0, 0) 2x(1y) = 0 2y = x2+1 (x, y) = (0, 1/2), (x, y) = (1, 1), (x, y) = (1, 1)

Hf(0, 1/2) =

1 0

0 2

(0, 1/2)

Hf(1, 1) =

0 2

2 2

Hf(0, 1/2) =

0 22 2

(1, 1)

(0, 0)

f(x, y) = (x+ 5y)(2x y)

10 0 5

f

(0, 0)

fx(x, y) = 4x y+ 10y fy(x, y) =x+ 10x 10y

fxx(x, y) = 4 fxy(x, y) =fyx(x, y) = 10 fyy(x, y) =10

fxx(0, 0) = 4 fxy(x, y) =fyx(0, 0) = 10 fyy(0, 0) =10

fxx(0, 0) > 0 (0, 0) f

>0

Hf(0, 0) =40 (10 )2 =2 20 100 =(+ 10)2 0


128/236

=10

f

(0, 0)

f(x, y) = (x+ 5y)(2x+ 10y) = 2(x+ 5y)2 0;

f(0, 0) = 0

=10

f(x, y) =ex2

(x2 + y2)

x >0 1< y


129/236

f(x, y) =x2 + 3y2 + 12 x

R2,

(x, y)R2 : x2 + 4 y2 4 ,

f(x, y) =

2x+1

2, 6y

f(x, y) = (0, 0)(x, y) =

14

, 0

.

14

, 0

f

fxx(x, y) = 2 fxy(x, y) =fyx(x, y) = 0 fyy(x, y) = 6

Hf

1

4, 0

=

2 0

0 6

12 > 0

fxx

14

, 0

= 2 > 0

14

, 0

f

E E

14

, 0

f

E

C(R2)

x2

4 +y2 1

x= 2 cos

y= sin


130/236


131/236

f(x, y) = (0, 0)

y(2x+y 1) = 0

x(x+ 2y 1) = 0.

(0, 0)

(0, 1)

(1, 0)

13

, 13

T

f

13

, 13

= 1

27

f(x, y) =xy(x + y 1)

f(x, y)

T

13

, 13

T

1)f(x, y) = 2x2 3xy+y2

Q= [1, 1] [1, 1]2)f(x, y) =x+y+xy

S={(x, y) :x0, y0, x2 +y2 1}

3)f(x, y) = 2x2 x4 2y2

S={(x, y) :x2 +y2 1}4)f(x, y) =xy +y2 yx A={(x, y) R2 : 0x1, 0yx}5)f(x, y) = (x 1)2y+ (y 2)2 4 E={(x, y) : 0y9 (x 1)2}6)f(x, y) =x2 +y2 2x+ 6y

E={(x, y) : 0x2, x 5y0}

7)g(x, y) =x2 y2

Q= [1, 1] [1, 1]8)f(x, y) = (x 1) exy R= [0, 3] [1, 0]9)f(x, y) = 3 y2 x x3 +y2

A={(x, y) R2 : 0x2, 0y1}

10)f(x, y) = 4 x y+ 4 x E={(x, y) R2 : 5x2 + 5 y2 6 x y 6 x+ 10 y+ 4 0}


132/236


133/236

11)maxTf(x, y) =f(1, 1) =f(2, 1) = log 5 minTf(x, y) =f(0, 0) = 012)maxCf= 3 =f(1, 0) minCf=

1/8 =f(

1/4, 0)

13)maxSf=e3 =f(1, 1, 1) =f(1, 1, 1) minSf=e13 =f132 , 0, 132 14)minR f(x, y) = 1 e2 =f(1,

e 1) maxR f(x, y) =1 =f(0, 0)

15)minQ f=1 =f(0, 0) maxQ f= 11 =f(1, 1) =f(1, 1)16)minA f(x, y) = arctan

83

=f

43

, 23

maxA f(x, y) = arctan 8 =f(0, 2)

17)maxA f= 12e1/2 =f

1

2, 1

2

minA f=12e1/2 =f

1

2, 1

2

18)minDf=1/4 =f(

5/2, 0) =f(0, 5/2),

maxDf= 6 =f(0, 0)

=f(5/2, 5/2) =f(5/2, 5/2)

M=

(x,y,z) R3 : x2 xy+y2 z= 1, x2 +y2 = 1

(0, 0, 0) R3

f(x,y,z) =x2 + y2 + z2

M

3/2

(1/

2, 1/2, 1/2)

(1/2, 1/2, 1/2)

(1, 0, 0) (0, 1, 0)

Q= [0, 1] [0, 1] :={(x, y) : 0x1,0

y

1

}

Q

a)f(x, y) =x2 + 3y2 xy yb)f(x, y) =

1

2x y

c)f(x, y) =ex+y


134/236


135/236

V1 ={(x, y) : x24 + y2

9 1}

V2 ={(x, y) : x2 xy+ y2 1} V1 V2

a)f(x, y) =xy

b)f(x, y) =x2 + 3y

A ={(x,y,z) : x2

+y

2

+z

2

1}

A

a)f(x,y,z) =xyz

b)f(x,y,z) =x+y z

f(x,y,z) =x2y2z2

S={(x,y,z) :x2 +y2 +z2 = 1}.

f(x,y,z) =ex2

+yz2

S={(x,y,z) :x2/4 +y2 + 3z2 1}.


136/236

f(x,y,z) =x2 + 5y2

1

2

xy

x2 + 4y2 40.

f(x, y) = xy

x2 +y2

f(x,y,z) = (x+y+z)2

x2 + 2y2 + 3z2 = 1.

f(x,y,z) =z2 xy 1 = 0

f(x, y) = (x2 +y2 +xy)2

x2 +y2 = 1.


137/236

f(x, y) = x2 + 3xy+y2

2x2 + 3xy+ 2y2

x2

16+

y2

9 = 1

f(x, y) =

x2 +y2 +y2 1

{(x, y) :x2 +y2 9}.

f(x, y) =|y 1|(2 y x2)

E={(x, y) R2 : 0< y


138/236

f(x, y) = log(1 +x+y+ y2 x)

D={(x, y) R2 :y < x < y2, y2}.

f(x, y) =ex2+y2 1

2x2 y2

D={(x, y) R2 : 3x2 + 4y2 4, y 1

2}.

f(x, y) =y2(x2 +y2 2x)

D={(x, y)R2 :x2 +y2


139/236

f(x, y) =x g(x, y) =y2 x3. (0, 0)

f

g(x, y) = 0,

f

f(0, 0) =g(0, 0).

f(x, y) = (y x2)3

E={(x, y)R

2

:x+ 2y

4 x2

}.

x2 + y2 z2 = 1 x + y+ 2z= 0;

f(x, y) = (y

x2)(x

y2) g(x, y) = y

x.

f

g= 0.

f

f(x, y) = (1 x2 4y2)2

Q={(x, y)R

2

:1x1, 1y1}.


140/236

f(x, y) =ex2y2

D=

(x, y)R2 : 3

x2 1y1 |x|

2

.

f(x, y) =x2 +y2 + 2x

C={(x, y) R2 :x2 + (y 2)2 1}.

m R,

f(x, y) =y mx

D=

(x, y) R2 :x0, y0, y3 x, y1 x4

.

f(x, y) = x2 y2

(2 +x2 +y2)2

S={(x, y) R2 :1y1}.


141/236

f(x, y) =ex22y

e2x

2y

E={(x, y) R2 :x0, y0}.

f(x, y) = (x3y2 +xy)ex2y

Da ={(x, y)R2 : 0xy}.

f(x, y) = sin(x+y) cos(x y)

S=

(x, y) R2 :|x| 2

, |y| 2

esercizi di analisi matematica 2

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Transcript of esercizi di analisi matematica 2